Problem 76 - Chemistry The Central Science

Question

A sample of \(3.00 \mathrm{~g}\) of \(\mathrm{SO}_{2}(g)\) originally in a \(5.00-\mathrm{L}\) vessel at \(21^{\circ} \mathrm{C}\) is transferred to a \(10.0-\mathrm{L}\) vessel at \(26^{\circ} \mathrm{C}\). A sample of \(2.35 \mathrm{~g} \mathrm{~N}_{2}(g)\) originally in a \(2.50-\mathrm{L}\) vessel at \(20{ }^{\circ} \mathrm{C}\) is transferred to this same 10.0 - \(\mathrm{L}\) vessel. (a) What is the partial pressure of \(\mathrm{SO}_{2}(g)\) in the larger container? (b) What is the partial pressure of \(\mathrm{N}_{2}(g)\) in this vessel? (c) What is the total pressure in the vessel?

Step-by-Step Solution

Answer

The partial pressure of SO₂(g) in the larger container is \(0.139 atm\), the partial pressure of N₂(g) is \(0.248 atm\), and the total pressure in the vessel is \(0.387 atm\).

Step 1: Find the moles of SO₂ and N₂ gas

To find the moles of each gas, we use the given mass of the gas and the molar mass of the gas. The molar mass of SO₂ is 32.07 g/mol (S) + 2 × 16 g/mol (O) = 64.07 g/mol. The molar mass of N₂ is 2 × 14 g/mol (N) = 28 g/mol. For SO₂: moles = mass / molar mass = 3 g / 64.07 g/mol = 0.0468 mol For N₂: moles = mass / molar mass = 2.35 g / 28 g/mol = 0.0839 mol

Step 2: Calculate initial pressures of both gases

Now that we have the number of moles for each gas, we can use the ideal gas equation to find the pressure of each gas in their initial containers. The ideal gas equation is PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant (0.08206 Latm/molK), and T is the temperature in Kelvin. First, convert the temperatures from Celsius to Kelvin: T₁ (SO₂) = 21°C + 273.15 = 294.15 K T₂ (N₂) = 20°C + 273.15 = 293.15 K Then, calculate the initial pressures: P₁ (SO₂) = (0.0468 mol × 0.08206 Latm/molK × 294.15 K) / 5.00 L = 0.251 atm P₂ (N₂) = (0.0839 mol × 0.08206 Latm/molK × 293.15 K) / 2.50 L = 0.906 atm

Step 3: Find new temperature and moles of each gas in the 10 L container

Temperature in the new 10 L container is 26°C - convert to Kelvin. T_new = 26°C + 273.15 = 299.15 K Now, we can use the ideal gas equation to find the new volume and partial pressures of the gases in the 10 L container using V₂ = V₁ × (T₂/T₁).

Step 4: Calculate final pressures of both gases

Since both gases are in the same 10 L container and at the same temperature (T_new), we can rearrange the ideal gas equation to solve for the final pressures of each gas. P_new (SO₂) = (0.0468 mol × 0.08206 Latm/molK × 299.15 K) / 10 L = 0.139 atm P_new (N₂) = (0.0839 mol × 0.08206 Latm/molK × 299.15 K) / 8.5 L = 0.248 atm

Step 5: Calculate total pressure inside the container

Now that we have the partial pressures of both gases, we can use Dalton's Law of partial pressures to find the total pressure inside the container: P_total = P_new (SO₂) + P_new (N₂). P_total = 0.139 atm (SO₂) + 0.248 atm (N₂) = 0.387 atm The partial pressure of SO₂(g) in the larger container is \(0.139 atm\), the partial pressure of N₂(g) is \(0.248 atm\), and the total pressure in the vessel is \(0.387 atm\).

Key Concepts

Partial Pressure

Partial pressure refers to the pressure that a single gas in a mixture contributes to the total pressure. In a container with mixed gases, each type of gas exerts its own pressure, known as its partial pressure. This is an essential concept, especially when dealing with reactions and processes involving gases in a mixed state. For instance, if a vessel contains both sulfur dioxide (SO₂) and nitrogen gas (N₂), the total pressure inside the vessel is the sum of the individual partial pressures of these gases.

The concept is mathematically expressed using the ideal gas law. For each gas in a mixture, you use its moles and the ideal gas equation: \[ P = \frac{{nRT}}{{V}} \] where:

• P is the partial pressure of the gas.
• n is the number of moles.
• R is the ideal gas constant.
• T is the temperature in Kelvin.
• V is the volume of the container.

This formula helps in calculating how much each gas contributes to the total pressure in the container. By understanding partial pressures, we can predict how the mixture will behave under different conditions.

Dalton's Law

Dalton's Law is a key principle in chemistry regarding gases. It states that the total pressure exerted by a mixture of non-reacting gases is the sum of their individual partial pressures. This means that each gas in a mixture behaves independently and the total pressure is simply the addition of the pressures each gas would exert if it were alone in the entire volume.

Understanding Dalton's Law is crucial for solving problems involving gas mixtures. The law is mathematically represented as:\[ P_\text{total} = P_1 + P_2 + ... + P_n \]where:

• \( P_\text{total} \) is the total pressure of the mixture.
• \( P_1, P_2, ..., P_n \) are the partial pressures of each individual gas.

This law is particularly useful when calculating the pressure within a container holding multiple gases, like in the exercise where both SO₂ and N₂ are present. Applying Dalton's Law helps determine how each gas contributes to the overall pressure, ensuring a clear understanding of the behavior of gas mixtures.

Gas Molar Mass

Gas molar mass is a fundamental concept in chemistry. It refers to the mass of one mole of a given gas and is typically measured in grams per mole (g/mol). Molar mass is crucial when converting the mass of a gas to moles, which is necessary for applying the ideal gas law and computing various properties of gases, such as pressure or volume.

To determine a gas's molar mass, you add up the atomic masses of the elements in the compound. For example:

• Sulfur dioxide (SO₂) has a molar mass of 64.07 g/mol, calculated as 32.07 g/mol (for S) + 2 × 16 g/mol (for O).
• Nitrogen gas (N₂) has a molar mass of 28 g/mol, derived from 2 × 14 g/mol (for N).

Understanding molar mass enables us to find the number of moles in a given sample, which is essential for using the ideal gas law to solve real-world problems. In our exercise, knowing the molar masses allows us to compute how many moles of SO₂ and N₂ are present, setting the stage for calculating their partial pressures.

Temperature Conversion

Temperature conversion is a pivotal step in calculations involving gas laws. Typically, the ideal gas law requires temperature input in Kelvin. Since many temperatures are given in degrees Celsius, conversion is necessary. This conversion ensures temperatures align with the absolute temperature scale required by gas equations.

To convert Celsius to Kelvin, use the following formula:\[ T(K) = T(°C) + 273.15 \]For instance:

• 21°C changes to 294.15 K.
• 26°C converts to 299.15 K.
• 20°C becomes 293.15 K.

Kelvin is used because it avoids negative values, simplifying computations involving gas laws. Consistently using Kelvin in calculations ensures accuracy, especially when applying the ideal gas law to determine properties like pressure or volume in gas mixtures.